Scott sentences for certain groups

Abstract

We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable $$\varSigma _3$$ Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are “computable d- $$\varSigma _2$$ ” (the conjunction of a computable $$\varSigma _2$$ sentence and a computable $$\varPi _2$$ sentence). In [9], this was shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group. Among the computable torsion-free abelian groups of finite rank, we focus on those of rank 1. These are exactly the additive subgroups of $$\mathbb {Q}$$ . We show that for some of these groups, the computable $$\varSigma _3$$ Scott sentence is best possible, while for others, there is a computable d- $$\varSigma _2$$ Scott sentence.